I want to create a lognormal distribution of future stock prices. Using a monte carlo simulation I came up with the standard deviation as being $\sqrt{(days/252)}$ $*volatility*mean*$ $\log(mean)$. Is this correct?

Thanks for the help, I figured it was a simple mistake. Actual formula for stdev is sqrt(days/252)*volatility*mean. The mistake I was making is days doesn't include first day. So if you have 5 days of random walk, days=4 in above formula.
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CptanPanicFeb 3 '11 at 1:34

3 Answers
3

I'm not sure I understand, but if you want to compute the variance of $exp(X)$, where $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, that variance is (from Wikipedia):
$$\left(\exp{(\sigma^2)} - 1\right) \exp{(2\mu + \sigma^2)}$$

I don't think I asked my question correctly. Assuming a random walk, starting at $\mu$ and yearly volatility of $\sigma$ . What is $\sigma$ for the distribution (expected prices) in like 1 month? I must be overthinking this, but it doesn't seem like $\sigma / 12 works either.
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CptanPanicFeb 1 '11 at 1:47

1

The volatility scales as the square root of time. So in one month, you would have $\sigma / \sqrt{12}$, not $\sigma / 12$. This has nothing to do with being log normal, though.
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shabbychefFeb 1 '11 at 3:29

The distribution of the log of a stock price in n days is a normal
distribution with mean of $\log(current_price)$ and standard deviation of
$volatility*\sqrt(n/365.2425)$ if you're using calendar days, and
assuming no dividends and 0% risk-free interest rate.

Note that the standard deviation is independent of the current_price:
if $\log(current_price)$ increases by 0.3 (for example), the stock has
increased by 35%, regardless of its current_price.